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11Lamina Flexural Response

11.1 Introduction

As defined in Chapters 8 through 10, pure, uniform tension, compression, and shear loadings must be individually applied to establish the fundamen-tal strength and stiffness properties of a composite material. A flexure test (i.e., the bending of a beam) typically induces tensile, compressive, and shear stresses simultaneously. Thus, this test is not usually a practical means of determining the fundamental properties of a composite material (Whitney et al. 1984, Adams 2000a).

Nevertheless, flexure tests are popular because of the simplicity of both specimen preparation and testing, as discussed subsequently. Gripping of the specimen, having the need for end tabs, obtaining a pure stress state, avoiding buckling, and most of the other concerns discussed in the previous three chapters are usually nonissues when conducting a flexure test.

Flexural testing can, for example, be a simple method of monitoring qual-ity during a structural fabrication process. The usual objective of a flexure test is to determine the flexural strength and flexural modulus of the beam material. This might be particularly relevant if the component being fabri-cated is to be subjected to flexural loading in service. However, because of the complex stress state present in the beam, it is typically not possible to directly relate the flexural properties obtained to the fundamental tensile, compressive, and shear properties of the material.

11.2 Testing Configurations

Figure11.1 indicates the configuration of the ASTM D790 (2010) three-point flexure test. This standard was created in 1970 by the plastics committee within the American Society for Testing and Materials (ASTM) for use with unreinforced and reinforced plastics and electrical insulating materials, as its title suggests. For more than 25 years, until 1996 when it was removed,

170 Experimental Characterization of Advanced Composite Materials

this standard also included four-point flexure. In response to demands by the composite materials community, a new standard, ASTM D6272 (2010), was introduced by the plastics committee specifying four-point flexure. That is, two standards now exist. The composites committee of ASTM has now written its own flexural test standard specifically for composite materials, ASTM D7264 (2007), which includes both three-point and four-point flexure. The three standards differ sufficiently in detail that it is advisable to refer to all three for guidance. It is unfortunate, but understandable because of its long existence, that many experimentalists still (incorrectly) only quote ASTM D790 as the governing standard for all flexural testing.

Analysis of a macroscopically homogeneous beam of linearly elastic material (Timoshenko 1984) shows that an applied bending moment is bal-anced by a linear distribution of normal stress x, as shown in Figure11.2. For the three-point flexural loading shown, the top surface of the beam is in compression, while the bottom surface is in tension. Assuming a beam of rectangular cross section, the midplane contains the neutral axis and is under zero bending stress. The shear force will be balanced by a distri-bution of interlaminar shear stress xz, varying parabolically from zero on the free surfaces to a maximum at the center as shown (Timoshenko 1984). For three-point flexure, the shear stress is constant along the length of the beam and directly proportional to the applied force P. However, the flex-ural stress, in addition to being directly proportional to P, is zero at each

z

P

P/2 P/2Shearstress, xz

Bendingstress, x

x

w

h

FIGURE 11.2Stresses in a beam subjected to three-point flexure.

h

P2

P2

L2

L

P

FIGURE 11.1Three-point flexure loading configuration.

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end support and maximum at the center. The stress state is highly depen-dent on the support span length-to-specimen thickness ratio (L/h). Beams with small L/h ratios are dominated by shear. As discussed in Chapter 10, a short-span (L/h = 4) three-point flexure test is used for interlaminar shear strength determination. Beams with long spans deflect more and usually fail in tension or compression. Typically, composite materials are stronger in tension than compression. Also, a concentrated load is applied at the point of maximum compressive stress in the beam, which tends to introduce local stress concentrations. Thus, the composite beam usually fails in compres-sion at the midspan loading point.

Although testing of a unidirectional composite is the primary subject of the present chapter, laminates of various other orientations can also be tested in flexure, as discussed in more detail in Chapter 15.

Flexural testing of a unidirectional composite is generally limited to beams with the fibers aligned parallel to the beam axis. That is, 0 flexural proper-ties are determined. Beams with fibers oriented perpendicular to the beam axis almost always fail in transverse tension on the lower surface because the transverse tensile strength of most composite materials is less than the trans-verse compressive strength, usually by a factor of three or more. In fact, the transverse flexure test has been suggested as a simple means of obtaining the transverse tensile strength of a unidirectional composite (Adams et al. 1990).

As do the other two flexural test standards, ASTM D7264 requires that a sufficiently large support span-to-specimen thickness ratio be chosen, such that failures occur in the outer fibers of the specimens, due only to the bend-ing moment. The standard recommends support span-to-thickness ratios of 16:1, 20:1, 32:1, 40:1, and 60:1, indicating that, as a general rule support span-to-thickness ratios of 16:1 are satisfactory when the ratio of the tensile strength to shear strength is less than 8 to 1. High-strength unidirectional composites can have much higher strength ratios, requiring correspondingly higher support span-to-thickness ratios. For example, ASTM D7264 suggests a ratio of 32:1 for three-point and four-point flexure. Although longer spans promote flexural failure, they also result in larger deflections. Excessive deflections lead to geometric nonlinear effects, which should be avoided.

Per ASTM D7264, the diameter of the loading noses and supports should be 6 mm. The other two ASTM standards have a slightly different require-ment, (viz., 10-mm diameters). This perhaps confirms experimental observa-tions that the test results obtained are not strongly influenced by the specific diameters used as long as the diameters are not so small that local bearing damage of the composite material occurs (Adams and Lewis 1995).

Four-point flexural loading is typically conducted in either third-point or quarter-point loading, as shown in Figure11.3. For quarter-point load-ing, the loading points are each positioned one-quarter of the support span length from the respective support and hence are one-half of the support span length from each other. For third-point loading, the loading points are each positioned one-third of the support span length from the respective

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support and hence are also one-third of the support span length from each other. ASTM D7264 includes only quarter-point loading.

Although the specimen thickness can be arbitrary (as long as the recom-mended span-to-thickness ratio is maintained), ASTM D7264 suggests a speci-men that is 100 mm long, 2.4 mm thick, and 13 mm wide and a 76.8-mm span for materials of high-strength. This results in a specimen overhang of 11.6 mm at each end. Suggestions are given for other types of composites as well.

11.3 Three- versus Four-Point Flexure

As noted in the previous section, either three- or four-point flexure can be, and is, used. The various ASTM standards make no specific recommenda-tions concerning when to use each test. In fact, there is no clear advantage of one test over the other, although there are significant differences. Figure11.4 indicates the required loadings and the corresponding bending moment M and transverse shear force V distributions in the beam for each of the loadings.

For three-point flexure, the maximum bending moment in the beam, and hence the location of the maximum tensile and compressive flexural stresses, is at midspan and is equal to = /4maxM PL . For four-point flexure with load-ing at the quarter points, attaining the same maximum bending moment, that is, = /4maxM PL , requires twice the testing machine force (i.e., 2P); this maximum bending moment is constant over the entire span L/2 between the applied loads (Figure11.4b). For four-point flexure with loading at the one-third points, attaining the same maximum bending moment, that is,

= =( /3) /4max 3 4M P L PL , requires 50% more testing machine force (i.e., 1.5P) than for three-point flexure. This maximum bending moment is constant over the entire span L/3 between the applied loads.

Thus, for quarter-point loading the force exerted by the testing machine must be twice as high, and for third-point loading the force must be one and one-half times as high, as for three-point flexure. Normally, this is not in itself a significant factor because the loads required to fail a beam in flexure are not extremely high. More significant is the magnitude of the required force at the loading point(s). It is realized that the highest flexural stresses

(b) ird-point loading(a) Quarter-point loading

L/2P P

hL

L/3

hL

FIGURE 11.3Four-point flexure test configurations: (a) quarter-point loading and (b) third-point loading.

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occur at these locations, and these stresses will be magnified by the stress concentration factors at the load introduction points. As noted in Figure11.4, for both three-point flexure and four-point flexure with quarter-point load-ing, the maximum concentrated force on the beam is P. However, for four-point flexure with third-point loading, the load applied is only 3 4 P . This indicates an advantage for third-point loading.

There are other considerations, however. The transverse shear force V, and hence the interlaminar shear stress in the beam, also varies with the type of loading, as indicated in Figure 11.4. For three-point loading, the shear force is equal to 12 P and is constant over the entire support span. For four-point flexure, quarter-point loading, the maximum shear force is equal to P and exists only over the end quarters of the beam. For four-point flexure, third-point loading, the maximum shear force is equal to 3 4 P and exists only over the end thirds of the beam. To avoid shear failure, it is desirable to keep the ratio of interlaminar shear stress to flexural stress sufficiently low. As explained, this is normally achieved by increasing the span length-to-specimen thickness ratio. For a given specimen thickness, three-point flex-ure ( )12V P= would be preferred because it minimizes the required support span length, followed by four-point flexure, third-point loading ( )3 4V P= .

One additional consideration should be noted, although it is usually of lesser importance. As discussed in Section 11.5, it may also be desired to determine the flexural modulus from the measured deflection of the beam.

ree-point flexure Four-point flexurethird-point loading

P

hL

P

Mmax = PL/4 Mmax = PL/4 Mmax = PL/4

(a)

(b)

V = P V = P V = P

(c)

P

L/2 PPh

LP P

L/3PPh

LP P

Four-point flexurequarter-point loading

FIGURE 11.4Required loadings for equal maximum bending moment in various beam configurations, with corresponding vertical shear force distributions: (a) loading diagrams, (b) moment diagrams, and (c) shear diagrams.

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The shear stresses in the beam contribute to the total deformation in pro-portion to the product of the shear force and the length over which it acts. The entire length of the beam in three-point flexure is subjected to the shear force 12 P, one-half of the four-point flexure; the quarter-point loading beam is subjected to the shear force P and two-thirds of the four-point flexure; and the third-point loading beam is subjected to the shear force 3 4 P. Hence, the net shear deformation is the same for all three cases. This consideration is important when beam deflection is used to determine modulus.

In summary, four-point flexure with third-point loading appears to be a good overall choice. However, each of the three loading modes (Figure11.4) has some individual advantages. Thus, all three are used, with three-point flexure the most common, perhaps only because it requires the simplest test fixture. A typical test fixture, with adjustable loading and support spans such that it can be used for both three- and four-point flexure, is shown in Figure11.5.

As an aside, note that three-point flexure is commonly (although perhaps erroneously) referred to as three-point loading, and likewise four-point

FIGURE 11.5Photograph of a flexure test fixture with interchangeable three- and four-point loading heads and adjustable spans. (Photograph courtesy of Wyoming Test Fixtures, Inc.)

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flexure is referred to as four-point loading. As a result, this has become accepted terminology, included in all three ASTM standards.

11.4 Specimen Preparation and Flexure Test Procedure

The flexure specimen is simply a strip of test material of constant width and thickness. As noted in Section 11.2, for a unidirectional [0]n composite under flexure, the suggested dimensions in ASTM D 7264 are support 76.8 mm span length, 100 mm specimen total length, 13 mm specimen width, and 2.4 mm specimen thickness. Suggested tolerances on these dimensions are also given in the standard.

Although all three ASTM standards specify the use of a deflection-measuring device mounted under the midspan of the specimen, occasionally a strain gage is used instead. One longitudinal strain gage can be mounted at the midspan on the tension side (bottom surface) of the specimen. The test fix-ture support span is to be set according to the beam thickness, specimen mate-rial properties, and fiber orientation, as discussed previously. ASTM D7264 specifies that the specimen is to be loaded at a crosshead rate of 1 mm/min, although a crosshead rate between 1 and 5 mm/min is common.

The beam deflection is measured using a calibrated linear variable dif-ferential transformer (LVDT) at the beam midspan. Alternatively, the beam displacement may be approximated as the travel of the testing machine crosshead if the components of this travel that are due to the machine com-pliance and to the indentations of the specimen at the loading and support points are subtracted. If a strain gage is used, the specimen is to be placed in the fixture with the strain gage on the tension side of the beam and centered at midspan. The strain or displacement readings may be recorded continu-ously or at discrete load intervals. If discrete data are recorded, the load and strain-displacement readings should be taken at small load intervals, with at least 25 points in the linear response region (so that an accurate flexural modulus can be determined). The total number of data points should be enough to accurately describe the complete beam response to failure.

11.5 Data Reduction

From classical beam theory (Timoshenko 1984), the tensile and compres-sive stresses at the surfaces of the beam at the location where the bending moment is a maximum is

=

( /2)max

maxM hI

(11.1)

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where Mmax is the maximum bending moment, h is the thickness of the beam, and = /123I wh is the moment of inertia of a beam of rectangular cross sec-tion, with w being the beam width.

For three-point flexure, for example, the maximum bending moment occurs at the midlength of the beam. The maximum value is given by

= /4maxM PL (11.2)

Substitution of Mmax and I into Equation (11.1) gives

2max 2

3PLwh

= (11.3)

This equation enables construction of a stressstrain plot if load versus strain has been recorded.

A reasonable approximation for most materials is that the modulus in ten-sion is the same as in compression. If a strain gage is used, the initial slope of the flexural stressstrain plot can be obtained using a linear least-square fit. Figures11.6 and 11.7 show typical stressstrain curves obtained from three-point flexure tests.

When the three-point flexure specimen is not strain gaged, the flexural modulus may be determined from a plot of load P versus center deflection as

4

3

3ELwh

Pf =

(11.4)

2000Carbon/epoxy, [0]12E1f = 128 GPa1600

1200

800

400

00 0.4 0.8Strain, %

1.2 1.6

Stress, M

Pa

X1f = 1880 MPa

31

1f = 1.55%

FIGURE 11.6Flexural stressstrain response of a [0]12 carbonepoxy test specimen.

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This relation, however, assumes that shear deformation is negligible. For [90]n beams (fibers perpendicular to the beam axis), shear deformation is gen-erally insignificant, and Equation (11.4) should be accurate. For [0]n beams, however, the results of Zweben (1990) shown in Figure 11.8 illustrate that certain unidirectional composites require relatively long support spans to

2000Carbon/PEEK (APC-2), [0]16

1600

1200Stress, M

Pa

800

400

00 0.2 0.4 0.6 0.8 1.0

Strain, %

31

E1f = 115 GPaX1f = 1632 MPa1f = 1.4%

FIGURE 11.7Flexural stressstrain response of a [0]16 carbonPEEK (polyetheretherketone) test specimen.

50Flexural modulus

Apparent flexural modulus

Kevlar 49/polyester, [0]n

Span-to-ickness Ratio, L/h

3

0 20 40 60 80 100 120 140

1

40

30

20

Mod

ulus, G

Pa

10

0

FIGURE 11.8Apparent flexural modulus of a [0]n Kevlar 49polyester specimen as a function of span-to-thickness ratio. (From Zweben, C., 1990, in Delaware Composites Design Encyclopedia, Vol. 1, Technomic, Lancaster, PA, pp. 4970).

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minimize the influence of shear deformation and thus produce an accurate flexural modulus.

If a flexural test is conducted for which the shear deformation component is significant, the following equation may be used to evaluate the flexural modulus:

65

3

3

2

213

EL

4wh1

h EL G

Pf

f= +

(11.5)

where G13 is the interlaminar shear modulus. The second term in the paren-theses of Equation (11.5) is a shear correction factor that may be significant for composites with a high axial modulus and a low interlaminar shear modu-lus. The flexural and shear moduli (Ef and G13, respectively), however, may not be known prior to the test. For use of Equation (11.5), the flexural modu-lus Ef may be replaced by the tensile modulus E1, and the out-of-plane shear modulus can be approximated as the in-plane shear modulus G12 if the fibers are oriented along the beam axis. When E1 and G12 are not known, the flex-ural modulus can be evaluated using Equation (11.4) for tests conducted on a slightly loaded beam over a range of span lengths until a constant value is achieved at long span lengths, as suggested in Figure11.8.

Additional details of data reduction, including for four-point flexure, can be found in the three ASTM flexural testing standards cited in this chapter.

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Lamina Flexural Response11.1Introduction11.2Testing Configurations11.3Three- versus Four-Point Flexure11.4Specimen Preparation and Flexure Test Procedure11.5Data Reduction

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